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Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow
Perturbations of minimizing movements and curves of maximal slope
Via della Ricerca Scientifica 1, Rome, 00133, Italy |
We modify the De Giorgi's minimizing movements scheme for a functional $φ$, by perturbing the dissipation term, and find a condition on the perturbations which ensures the convergence of the scheme to an absolutely continuous perturbed minimizing movement. The perturbations produce a variation of the metric derivative of the minimizing movement. This process is formalized by the introduction of the notion of curve of maximal slope for $φ$ with a given rate. We show that if we relax the condition on the perturbations we may have many different meaningful effects; in particular, some perturbed minimizing movements may explore different potential wells.
References:
[1] |
N. Ansini, Gradient flows with wiggly potential: a variational approach to the dynamics, in
Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, CoMFoS16
(Springer), 30 (2017), 139-151.
doi: 10.1007/978-981-10-6283-4_12. |
[2] |
N. Ansini, A. Braides and J. Zimmer, Minimising movements for oscillating energies: The critical regime, Proceedings of the Royal Society of Edinburgh Section A (in press), 2016, arXiv: 1605.01885. |
[3] |
F. Almgren, J. E. Taylor and L. Wang,
Curvature-driven flows: a variational approach, SIAM Journal of Control and Optimization, 31 (1993), 387-438.
doi: 10.1137/0331020. |
[4] |
L. Ambrosio, N. Gigli and G. Savaré,
Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhauser Verlag, second edition, 2008. |
[5] |
A. Braides,
Γ-convergence for Beginners, Oxford University Press, 2002.
doi: 10.1093/acprof:oso/9780198507840.001.0001. |
[6] |
A. Braides,
Local Minimization, Variational Evolution and Γ-convergence, Springer Cham, 2014.
doi: 10.1007/978-3-319-01982-6. |
[7] |
A. Braides, M. Colombo, M. Gobbino and M. Solci,
Minimizing movements along a sequence of functionals and curves of maximal slope, Comptes Rendus Matematique, 354 (2005), 685-689.
doi: 10.1016/j.crma.2016.04.011. |
[8] |
M. Colombo and M. Gobbino, Passing to the limit in maximal slope curves: from a regularized Perona-Malik to the total variation flow,
Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250017, 19pp.
doi: 10.1142/S0218202512500170. |
[9] |
L. C. Evans,
Partial Differential Equations, American Mathematical Society, second edition, 2010.
doi: 10.1090/gsm/019. |
[10] |
F. Fleissner,
Γ-convergence and relaxations for gradient flows in metric spaces: A minimizing movement approach, ESAIM Control, Optimisation and Calculus of Variations (to appear), preprint, arXiv: 1603.02822, (2016). |
[11] |
F. Fleissner and G. Savaré, Reverse approximation of gradient flows as minimizing movements: A conjecture by De Giorgi,
Forthcoming Articles, (2018), 30pp, arXiv: 1711.07256.
doi: 10.2422/2036-2145.201711_008. |
[12] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Plank equation, SIAM Journal of Mathematical Analysis, 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[13] |
E. Sandier and S. Serfaty,
Gamma-convergence of gradient flows and application to Ginzburg-Landau, Communications on Pure and Applied Mathematics, 57 (2004), 1627-1672.
doi: 10.1002/cpa.20046. |
show all references
References:
[1] |
N. Ansini, Gradient flows with wiggly potential: a variational approach to the dynamics, in
Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, CoMFoS16
(Springer), 30 (2017), 139-151.
doi: 10.1007/978-981-10-6283-4_12. |
[2] |
N. Ansini, A. Braides and J. Zimmer, Minimising movements for oscillating energies: The critical regime, Proceedings of the Royal Society of Edinburgh Section A (in press), 2016, arXiv: 1605.01885. |
[3] |
F. Almgren, J. E. Taylor and L. Wang,
Curvature-driven flows: a variational approach, SIAM Journal of Control and Optimization, 31 (1993), 387-438.
doi: 10.1137/0331020. |
[4] |
L. Ambrosio, N. Gigli and G. Savaré,
Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhauser Verlag, second edition, 2008. |
[5] |
A. Braides,
Γ-convergence for Beginners, Oxford University Press, 2002.
doi: 10.1093/acprof:oso/9780198507840.001.0001. |
[6] |
A. Braides,
Local Minimization, Variational Evolution and Γ-convergence, Springer Cham, 2014.
doi: 10.1007/978-3-319-01982-6. |
[7] |
A. Braides, M. Colombo, M. Gobbino and M. Solci,
Minimizing movements along a sequence of functionals and curves of maximal slope, Comptes Rendus Matematique, 354 (2005), 685-689.
doi: 10.1016/j.crma.2016.04.011. |
[8] |
M. Colombo and M. Gobbino, Passing to the limit in maximal slope curves: from a regularized Perona-Malik to the total variation flow,
Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250017, 19pp.
doi: 10.1142/S0218202512500170. |
[9] |
L. C. Evans,
Partial Differential Equations, American Mathematical Society, second edition, 2010.
doi: 10.1090/gsm/019. |
[10] |
F. Fleissner,
Γ-convergence and relaxations for gradient flows in metric spaces: A minimizing movement approach, ESAIM Control, Optimisation and Calculus of Variations (to appear), preprint, arXiv: 1603.02822, (2016). |
[11] |
F. Fleissner and G. Savaré, Reverse approximation of gradient flows as minimizing movements: A conjecture by De Giorgi,
Forthcoming Articles, (2018), 30pp, arXiv: 1711.07256.
doi: 10.2422/2036-2145.201711_008. |
[12] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Plank equation, SIAM Journal of Mathematical Analysis, 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
[13] |
E. Sandier and S. Serfaty,
Gamma-convergence of gradient flows and application to Ginzburg-Landau, Communications on Pure and Applied Mathematics, 57 (2004), 1627-1672.
doi: 10.1002/cpa.20046. |






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